class: center, middle, inverse, title-slide .title[ # Class 2a: Review of concepts in Probability and Statistics ] .author[ ### Business Forecasting ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; } </style> --- ### Summary - In the last class: - We discussed the organization of the course - We overviewed forecasting methods - We learned about methods of qualitative forecasting - *Reference:* Forecasting Methods and Applications, chapter 1 - This set of classes: - We will start learning about exploratory analysis preparing the forecast - We will learn to translate business problems to statistical problems - We will learn about various **data types** - We will learn how to **summarize data and observed patterns** - *Reference:* Forecasting Methods and Applications, chapter 2.1-2.4 --- #### Why do we care? - To use scientific methods we need to set business problem as a statistical problem - Define thins that we want to find to identify appropriate methods --- ### Specific Scenario **The Opportunity** - Online pharmacies are rapidly growing, offering convenience and accessibility to customers. - Example in Mexico: *Choiz*, which provides prescription services and drug subscriptions. -- **The Trend** - A new generation of highly effective anti-diabetes medications, like *Ozempic*, is gaining popularity for dual benefits: managing diabetes and aiding weight loss. --- **Search Interest in Ozempic**  --- **Your Role** - You have been hired as a consultant for a start-up aiming to launch a subscription service for these medications in Mexico. <img src="data:image/png;base64,#Ozempic_ad.jpg" alt="Ozempic Advertisement" style="width:50%;"/> -- **The Challenge** - Your boss needs a detailed exploratory market analysis to forecast sales and identify key customer segments --- ### Parameters vs Statistics - You need to know how many people in Mexico have diabetes -- #### Parameter - Call `\(\mu_d\)` the proportion of Mexican population which has diabetes - Usually the parameter is an **unknown** number **describing the whole population** - You want to learn what it is - In our example, `\(\mu_d\)` is a parameter that you want to learn - More generally, parameter describes an aspect of the entire population -- #### Statistic - But you don't have data on the whole population. At best you can get a sample from a survey - So you will try to estimate this parameter with sample - Statistic is a **guess of the parameter** which can be **calculated from the sample** - You will calculate a statistic `\(\hat{\mu}_p\)` which is the proportion of diabetics in the sample --- ### Sampling: categorical variable  --- ### Sampling: continuous variable <!-- --> --- ### Parameters vs Statistics - What is population, sample, parameter and statistic in the following examples? -- - You want to know the probability that a user who got a match on tinder will go out on a date with that person. You survey 1000 users and ask them about each match they got if they went on a date. You then calculate the share of dates which ended up in a match for these users. -- - You want to know what whether starbucks baristas are faster than Cielito Querido baristas. You go to 10 starbucks and 10 Cielito Querido and measure the time it takes to make a coffee. You then calculate the average time it takes to make a coffee in each of these chains. -- - You want to know the average age of people who go to the gym. You go to a gym and ask 100 people about their age. You then calculate the average age of these people. -- - You want to know the variance of internet speed during in Mexico City. You visit 500 households and calculate the variance of their internet speed. --- ### Parameters vs Statistics - What are some examples of the statistics we will use and why do we care? --- layout: false class: inverse, middle # Summary Statistics --- ### Summary Statistics - Any advanced analysis always starts with describing the data 1. **Cambridge Analytica** - Analyzed voter demographics and turnout patterns to target swing voters with tailored political ads. -- 2. **Tesla** - Summarized battery efficiency and charging times to improve EV performance and user experience. -- 3. **Walmart** - Optimized opening hours and staffing by analyzing distribution -- 4. **BBVA** - Analyzed average credit card balances and repayment trends to design tailored credit offers and improve customer retention. -- Basically any modern company relies on summarizing data to make decisions --- # Measures of Central Tendency ## Mean - **Mean** represents the arithmetic average of the data. - Sometimes called the expected value of the random variable E(X) - The population mean `\(\mu\)` is the sum of all observations divided by the total population size: `$$\mu =E(X)=\frac{\sum_{i=1}^{N} x_i}{N}=\sum_{x\in X}P(X=x) \times x$$` - where `\(N\)` is the total population size, and `\(x_i\)` are individual data points. - The sample mean, denoted as `\(\bar{x}\)`, is the sample equivalent: `$$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1+x_2+...x_{n-1}+x_n}{n}$$` where `\(n\)` is the sample size. --- ## Mean Intuitively, mean is the balancing point of the distribution. <!-- --> --- ## Mean of a binary variable What if a mean of a **binary variable**? - Binary variable is a variable which takes value 0 or 1 - For example: do you have diabetes (yes=1, no=0) -- What is the intuitive interpretation of the mean of this variable? - `\(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)` - `\(\bar{x} = \frac{1+0+0+...0+1}{n}=\frac{n_{diabetes}}{n}=\hat{\mu}_{diabetes}\)` -- It's the proportion of people with diabetes in the sample: mean(diabetes)= 0.11 --- ## Weighted Mean - In some scenarios, data points have different weights. - For a dataset with weights `\(w_i\)` and values `\(x_i\)`, the weighted mean is: `$$\small \text{Weighted Mean} = \frac{\sum_{i=1}^{n} w_i \cdot x_i}{\sum_{i=1}^{n} w_i}$$` <div class="datatables html-widget html-fill-item" id="htmlwidget-b7613d44a90b027911f1" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-b7613d44a90b027911f1">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["Midterm 1","Midterm 2","Quizzes","Final Project","Final exam"],[0.2,0.2,0.15,0.15,0.3],[6,8,9,4,8]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Person<\/th>\n <th>Weight<\/th>\n <th>Grade<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":4,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]},{"name":"Person","targets":0},{"name":"Weight","targets":1},{"name":"Grade","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> The ** weighted mean** is: `\begin{align*} \small \bar{x} & =\frac{0.2\times 6+0.2\times 8+0.15 \times 9+ 0.15 \times 4+0.3 \times 8}{0.2+0.2+0.15+0.15+0.3} \end{align*}` --- ## Mean - Is mean always a right measure? #### "Bill Gates walks into a bar" - Suppose a group of people, including Bill Gates, walks into a bar. - Let's say the net worth of everyone in the group is as follows: .pull-left[ <div class="datatables html-widget html-fill-item" id="htmlwidget-b1aec08772e71d9332a1" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-b1aec08772e71d9332a1">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["Person 1","Person 2","Person 3","Person 4","Person 5","Bill Gates"],[10,20,30,40,50,600000]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Person<\/th>\n <th>Net_Worth<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":1},{"name":"Person","targets":0},{"name":"Net_Worth","targets":1}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> ] .pull-right[ The **mean** is: `\begin{align*} \bar{x} & =\frac{10 + 20 + 30 + 40 + 50 + 60000}{6} \\ & = 10025 \\ \end{align*}` Mean is seriously skewed due to the outlier. ] --- ## Mean vs Median <center> <img src=mean_median.jpg width="800"> </center> --- ## Median - **Median** represents the middle value when data is sorted - Half of observations are below it, half are above it. - For a dataset with odd size `\(n\)`, the median is the `\(\frac{n+1}{2}\)`-th value - For even size `\(n\)`, it's the average of `\(\frac{n}{2}\)`-th and `\(\frac{n}{2}+1\)`-th values. .pull-left[ | Day | Number of Customers | |-----|---------------------| | 1 | 20 | | 2 | 18 | | 3 | 25 | | 4 | 22 | | 5 | 30 | | 6 | 21 | | 7 | 27 | ] .pull-right[ The dataset has `\(n=7\)` (odd) observations, so to find the median: - Arrange the data in ascending order: - 18, 20, 21, 22, 25, 27, 30. - The median is the `\(\frac{n+1}{2}\)`-th value, which is the 4th value. - Thus, the median is the 4th value, which is 22. ] --- ### Let's look at the median weight in our population - Mean: 72.66451 - Median: 70.7536 -- <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/unnamed-chunk-6-1.png" width="100%" /> - Mean is dotted - Median is dashed --- ### Median and outliers I added couple of observations on the right tail of the distribution - Old Mean: 72.66, **New Mean: 77.05** - Old Median: 70.75, **New Median: 70.95** <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/abc-1.png" width="100%" /> --- ## Side note on the Mode **Mode** is the most frequent value in the data - Let's look at the distribution of age of people with diabtese <div class="datatables html-widget html-fill-item" id="htmlwidget-bf14b3fe527cf9eaee51" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-bf14b3fe527cf9eaee51">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97"],[4,2,4,3,5,7,6,6,5,12,16,14,16,15,33,33,31,32,48,61,37,47,82,73,65,75,67,85,64,114,98,87,115,99,105,103,124,100,122,119,133,99,114,115,107,129,112,112,116,96,90,74,82,91,69,73,59,45,51,46,45,17,41,23,25,18,13,13,10,5,4,1,4,1,2,1,1,1],[0.001,0,0.001,0.001,0.001,0.002,0.001,0.001,0.001,0.003,0.004,0.003,0.004,0.004,0.008,0.008,0.007,0.008,0.012,0.015,0.008999999999999999,0.011,0.02,0.017,0.016,0.018,0.016,0.02,0.015,0.027,0.023,0.021,0.028,0.024,0.025,0.025,0.03,0.024,0.029,0.029,0.032,0.024,0.027,0.028,0.026,0.031,0.027,0.027,0.028,0.023,0.022,0.018,0.02,0.022,0.017,0.017,0.014,0.011,0.012,0.011,0.011,0.004,0.01,0.006,0.006,0.004,0.003,0.003,0.002,0.001,0.001,0,0.001,0,0,0,0,0]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Age<\/th>\n <th>n_i<\/th>\n <th>p_i<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]},{"name":"Age","targets":0},{"name":"n_i","targets":1},{"name":"p_i","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ## Mode <center> <img src=mode.jpg width="400"> </center> --- ## Percentiles <center> <img src=Natalie.jpg width="300"> </center> --- ## Percentiles <center> <img src=t_swift2.jpg width="300"> </center> --- ## Percentiles <center> <img src=t_swift_01.jpg width="300"> </center> --- --- ## Percentiles <center> <img src=t_swift_0001.jpg width="300"> </center> --- ## Percentiles <center> <img src=WSJ.jpg width="800"> </center> Credits: Wall Street Journal [Article Link](https://www.wsj.com/tech/personal-tech/spotify-wrapped-2023-taylor-swift-e303333d) --- ## Percentiles - How much inventory of milk you need to keep in your Starbucks? -- - What is the tradeoff of keeping too much vs too litle inventory? -- - Suppose we want to have enough of milk to cover sales on 95% of days -- - To figure it out, let's look at the distribution of the daily use of milk <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/Sales_dist_figure-1.png" width="100%" /> --- ## Percentiles - Let `\(s_i\)` be the daily sales of milk - We want to choose amount `\(M\)`, such that `\(P(s_i \leq M)=0.95\)` - That is, in 95% of days sales are smaller or equal than `\(M\)` -- <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/Sales_dist_figure_with_shaded_region-1.png" width="100%" /> -- - What is this number? - It's the 95th percentile of the distribution (274 liters) --- ## Percentiles - *Percentiles* divide the ordered data into 100 equal parts. - `\(p\)`th percentile is a value such that `\(p\%\)` of the data are below it - `\(v_p\)` is such that `\(P(x_i \leq v_p)=p\)` - `\(v_{95}\)` is such that `\(P(x_i \leq v_{95})=95\%\)` --- ## Percentiles -- - What is the the height such that 75% of ITAM students are smaller than this height? -- - What is the income level such that 25% of people in Mexico earn less than that level? -- - What is the age, such that 50% of people die before that age? -- - When do I need to leave home to be on time for class 90% of the days? --- ## How to find it in a sample 1. Arrange the data in ascending order -- 2. Find which observation corresponds to the relevant percentile - Formula: `\(i = \left(\frac{p}{100}\right)(n+1)\)` - Example: To find 95th percentile in a sample of 1000 observations we look at `\(i = \left(\frac{95}{100}\right)(1000+1)=950.95\)` observation -- 3. If it's an integer, value of ith observation is your percentile 4. If it's not, take the average between ith rounded down and ith rounded up - In our example it would be the average of 950th and 951th observation --- ## Or use the CDF - `\(ECDF(v)=P(x_i \leq v)\)` <div id="htmlwidget-2717a99ca11583d690eb" style="width:800px;height:300px;" class="plotly html-widget"></div> <script type="application/json" 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","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ## Common values - **Median** - 50th percentile - half of the values are below the median - **Quartiles** - 25th, 50th and 75th percentile. - How poor is the poorest quartile of the society? - Their income is below the 25th percentile <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/Image_with_shaded_area_Qtiles-1.png" width="100%" /> --- - **Deciles** - 10th, 20th, ... 90th - How bad pollution gets in CDMX during top 10% polluted days? - During top 10% of polluted days pollution level is larger or than 9th decile. <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/Image_with_shaded_area_Deciles-1.png" width="100%" /> --- ### Example with data Here is a data on distribution of how many views have various tik-tok videos. - What is the 1st decile? - What is the 95th percentile? <div class="datatables html-widget html-fill-item" id="htmlwidget-7693b226083ebd1214ea" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-7693b226083ebd1214ea">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["TikTok Video 1","TikTok Video 2","TikTok Video 3","TikTok Video 4","TikTok Video 5","TikTok Video 6","TikTok Video 7","TikTok Video 8","TikTok Video 9","TikTok Video 10","TikTok Video 11","TikTok Video 12","TikTok Video 13","TikTok Video 14","TikTok Video 15","TikTok Video 16","TikTok Video 17","TikTok Video 18","TikTok Video 19","TikTok Video 20","TikTok Video 21","TikTok Video 22","TikTok Video 23","TikTok Video 24","TikTok Video 25","TikTok Video 26","TikTok Video 27","TikTok Video 28","TikTok Video 29","TikTok Video 30","TikTok Video 31","TikTok Video 32","TikTok Video 33","TikTok Video 34","TikTok Video 35","TikTok Video 36","TikTok Video 37","TikTok Video 38","TikTok Video 39","TikTok Video 40","TikTok Video 41","TikTok Video 42","TikTok Video 43","TikTok Video 44","TikTok Video 45","TikTok Video 46","TikTok Video 47","TikTok Video 48","TikTok Video 49","TikTok Video 50","TikTok Video 51","TikTok Video 52","TikTok Video 53","TikTok Video 54","TikTok Video 55","TikTok Video 56","TikTok Video 57","TikTok Video 58","TikTok Video 59","TikTok Video 60","TikTok Video 61","TikTok Video 62","TikTok Video 63","TikTok Video 64","TikTok Video 65","TikTok Video 66","TikTok Video 67","TikTok Video 68","TikTok Video 69","TikTok Video 70","TikTok Video 71","TikTok Video 72","TikTok Video 73","TikTok Video 74","TikTok Video 75","TikTok Video 76","TikTok Video 77","TikTok Video 78","TikTok Video 79","TikTok Video 80","TikTok Video 81","TikTok Video 82","TikTok Video 83","TikTok Video 84","TikTok Video 85","TikTok Video 86","TikTok Video 87","TikTok Video 88","TikTok Video 89","TikTok Video 90","TikTok Video 91","TikTok Video 92","TikTok Video 93","TikTok Video 94","TikTok Video 95","TikTok Video 96","TikTok Video 97","TikTok Video 98","TikTok Video 99","TikTok Video 100","TikTok Video 101","TikTok Video 102","TikTok Video 103","TikTok Video 104","TikTok Video 105","TikTok Video 106","TikTok Video 107","TikTok Video 108","TikTok Video 109","TikTok Video 110","TikTok Video 111","TikTok Video 112","TikTok Video 113","TikTok Video 114","TikTok Video 115","TikTok Video 116","TikTok Video 117","TikTok Video 118","TikTok Video 119","TikTok Video 120","TikTok Video 121","TikTok Video 122","TikTok Video 123","TikTok Video 124","TikTok Video 125","TikTok Video 126","TikTok Video 127","TikTok Video 128","TikTok Video 129","TikTok Video 130","TikTok Video 131","TikTok Video 132","TikTok Video 133","TikTok Video 134","TikTok Video 135","TikTok Video 136","TikTok Video 137","TikTok Video 138","TikTok Video 139","TikTok Video 140","TikTok Video 141","TikTok Video 142","TikTok Video 143","TikTok Video 144","TikTok Video 145","TikTok Video 146","TikTok Video 147","TikTok Video 148","TikTok Video 149","TikTok Video 150","TikTok Video 151","TikTok Video 152","TikTok Video 153","TikTok Video 154","TikTok Video 155","TikTok Video 156","TikTok Video 157","TikTok Video 158","TikTok Video 159","TikTok Video 160","TikTok Video 161","TikTok Video 162","TikTok Video 163","TikTok Video 164","TikTok Video 165","TikTok Video 166","TikTok Video 167","TikTok Video 168","TikTok Video 169","TikTok Video 170","TikTok Video 171","TikTok Video 172","TikTok Video 173","TikTok Video 174","TikTok Video 175","TikTok Video 176","TikTok Video 177","TikTok Video 178","TikTok Video 179","TikTok Video 180","TikTok Video 181","TikTok Video 182","TikTok Video 183","TikTok Video 184","TikTok Video 185","TikTok Video 186","TikTok Video 187","TikTok Video 188","TikTok Video 189","TikTok Video 190","TikTok Video 191","TikTok Video 192","TikTok Video 193","TikTok Video 194","TikTok Video 195","TikTok Video 196","TikTok Video 197","TikTok Video 198","TikTok Video 199","TikTok Video 200"],[172204,9442,37975,56914,40393,18785,212625,19111,454810,20049,155952,680082,2743,14499,18034,57178,14380,410,566,159559,13906,1522,17020,136218,378053,11548,14974,1564,43921,8434,43616,63402,104053,8836,46978,1677,6791,6147,589,23253,30001,12816,68682,7405,2829,42160,6522,192173,11531,58893,35699,6797,234121,57777,25201,33350,61019,25204,247,33770,12697,29081,52719,179801,7399,155409,36453,104585,87649,64946,4607,19239,56122,5270,9757,52654,69722,44163,5833,4232,213003,32431,25151,18373,3672,55160,15903,16745,89323,75558,177757,10783,58426,177489,4162,6056,4034,2468,24834,58677,133445,105570,4891,352471,8102,25804,11691,18333,29211,26337,21213,25903,10634,10339,1823,12413,10209,1267854,2855,27062,2344,2427,26557,4940,21966,11587,8774,1057,3508,28833,51608,10516,22029,118698,190953,4249,18472,133552,10888,20359,19358,5817,11305,21075,11840,117018,10705,11502,62648,4516,20722,2149,126849,14611,10919,3438,21771,6631,9895,151981,16928,4413,28136,12783,53371,188836,4969,43564,25018,84402,15605,77260,1607,277670,80593,17568,2506,57787,45469,21817,27645,9171,38300,34268,14489,2968,63015,50579,6283,2014,29955,13126,32174,3162,5225,112270,40363,53089,335321,26722]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>VideoTitle<\/th>\n <th>Views<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":4,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":1},{"name":"VideoTitle","targets":0},{"name":"Views","targets":1}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- - Index for the first decile is: `\(i = \left(\frac{10}{100}\right)(200+1)=20.1\)` - First decile is the average of the 20th and 21st observation - Index for the 95th percentile is: `\(i = \left(\frac{95}{100}\right)(200+1)=190.95\)` - 95th percentile is the average of the at 190th and 191st observation --- --- # Measures of Dispersion -- - Suppose a store has an average daily revenue of 10 000 pesos - It could be that on each day it has exactly 10 000 pesos revenue - Or it could be that on half of days it gets 20 000 pesos but on other half it gets 0 pesos - Dispersion makes a big difference, especially when trying to understand risk! -- ## Range - **Range** the difference between minimum and maximum value in the data `$$R=x_{max}-x_{min}$$` -- - What is the difference between the oldest and the youngest person with diabetes? - **R**=77=97-20 --- - Very sensitive to outliers <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/11-1.png" width="100%" /> --- ## Interquartile Range - **Interquartile range** is the difference between the first and the third quartile of the data: `$$IQR=q_3-q_1$$` - What is the IQR of age in people with diabetes? - **IQR**=19=68-49 -- - 50% of the sample is between `\(q_3\)` and `\(q_1\)` -- <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/2-1.png" width="100%" /> --- - Is it more or less sensitive to outliers than range? <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/1-1.png" width="100%" /> --- ### Interquartile Range --- ### Example with data - What is the IQR? <div class="datatables html-widget html-fill-item" id="htmlwidget-6abeebd0617386b043ac" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-6abeebd0617386b043ac">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["TikTok Video 1","TikTok Video 2","TikTok Video 3","TikTok Video 4","TikTok Video 5","TikTok Video 6","TikTok Video 7","TikTok Video 8","TikTok Video 9","TikTok Video 10","TikTok Video 11","TikTok Video 12","TikTok Video 13","TikTok Video 14","TikTok Video 15","TikTok Video 16","TikTok Video 17","TikTok Video 18","TikTok Video 19","TikTok Video 20"],[30,17,22,24,23,19,32,20,37,20,30,40,13,18,19,24,18,9,10,30]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>VideoTitle<\/th>\n <th>Views<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":4,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":1},{"name":"VideoTitle","targets":0},{"name":"Views","targets":1}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### Example with data Here is a (smaller) data on distribution of how many views have various tik-tok videos. - Suppose that all views triples and 1000 additional people viewed them as well `$$y_i=3x_i+1000$$` - What is new IQR? <div class="datatables html-widget html-fill-item" id="htmlwidget-72405b50a3b8a19adc99" style="width:100%;height:auto;"></div> <script type="application/json" 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'12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]},{"name":"VideoTitle","targets":0},{"name":"OldViews","targets":1},{"name":"NewViews","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### IQR - Order of observations was not affected, so same observations correspond to the first and the third quartile `$$q^{New}_{1}=3q^{Old}_1+1000$$` `$$q^{New}_{3}=3q^{Old}_3+1000$$` -- - And more generally, for `$$y_i=bx_i+a$$` and `\(b>0\)` `$$v^y_p=bv^{x}_p+a$$` - if `\(b<0\)` then the order reverses. -- - So what does it mean for IQR? `$$IQR^{New}=q^{New}_{3}-q^{New}_{1}=3q^{Old}_3-3q^{Old}_1=3*IQR^{Old}$$` --- <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/5aa-1.png" width="100%" /> --- ### Variance & Standard Deviation **Variance** measures how far an average observation is from the mean: - **Population variance**: `$$\sigma^2=E[(X-\mu)^2]=E[X^2]-\mu^2 =\frac{1}{N} \sum^N_{i=1}(x_i-\mu)^2=\frac{1}{N} (\sum^N_{i=1}x^2_i-N\mu^2)$$` For Discrete Variables it can be: `\(\sigma^2=\sum_k P(X=k)(k-\mu)^2\)`, where k is any possible value that X can take. -- - But variance does not have the right units since it squares everything... -- - **Population standard deviation** deviation: `$$\sigma=\sqrt{\frac{1}{N} \sum^N_{i=1}(x_i-\mu)^2}$$` --- ### Variance & Standard Deviation -- - Why do we first take squares and then take square root? -- - Can't we just do `\(\frac{1}{N}\sum^N_{i=1}(x_i-\mu)\)`? -- - NO! Because `\(\\sum^N_{i=1}(x_i-\mu)=0\)` -- - Why don't we just do Mean Absolute Deviation? - `\(MAD=frac{1}{N}\sum^N_{i=1}|(x_i-\mu)|\)` -- - MAD is not differentiable at 0 :( -- - Variance puts more weight on far away observations. - It's a weighted average distance, where weights are distances themselves. -- `$$\sigma^2=\sum^N_{i=1}(x_i-\mu)^2=\frac{1}{N}\sum^N_{i=1} \underbrace{|(x_i-\mu)|}_{Weight}*\underbrace{|(x_i-\mu)|}_{Distance}$$` - It makes it very sensitive to outliers, which get a lot of weight! - Standard deviation retains this property --- <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/3-1.png" width="100%" /> --- <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/3b-1.png" width="100%" /> --- ### Variance & Standard Deviation Consider two bets/situations: - Bet A: with 75% you get 200 pesos and with 25% you lose me 200 pesos - 75% chance your life goes normal and you keep making money - 25% change your house burns down -- - Bet B: with 75% you get 110 pesos and with 25% you get 70 - When your life goes normal you get 110 (you pay 90 for insurance) - When your house burns down you are paid some compensation (70) -- - Compute expected value and variance of each bet - Which one would you prefer? -- - What if I change Bet B: - Bet B: with 75% you get 109 pesos and with 25% you get 69 -- - That's how insurance companies make profits --- ### Sample equivalents - **Sample variance**: `$$s^2=\frac{1}{n-1} \sum^n_{i=1}(x_i-\bar{x})^2=\frac{1}{n-1}( \sum^n_{i=1}x_i^2-n\bar{x}^2)$$` -- - **Sample standard deviation** deviation: `$$s=\sqrt{\frac{1}{n-1}( \sum^n_{i=1}x_i-\bar{x})^2}=\sqrt{\frac{1}{n-1}( \sum^n_{i=1}x_i^2-n\bar{x}^2)}$$` -- - Why we divide by `\(n-1\)` rather than `\(n\)`? -- --- - **Intuition** - observed values usually fall closer to the sample mean than to the population mean. Distances are artificially small. <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/4-1.png" width="100%" /> --- ### Sample equivalents - So the deviations from the sample mean underestimate the population standard deviation -- - So we divide by a smaller number to correct for it -- - In big sample `\(\frac{1}{n}\)` and `\(\frac{1}{n-1}\)` are similar, so correction doesn't matter as much --- - **Intuition** - in big samples, our estimate of the population mean is already good, no need to correct <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/4b-1.png" width="100%" /> --- ### Standard Deviation Consider a random variable X with `\(E(X)=\mu_x\)` and standard deviation `\(\sigma_x\)`. Ex: X is number of instagram followers distributed like this: <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/5aag-1.png" width="100%" /> --- ### Standard Deviation - What happens to mean and standard deviation if everyone gets 40 more followers? - `\(Y=X+40\)` <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/5aagfb-1.png" width="100%" /> -- `$$\scriptsize \mu_Y=E(Y)=\frac{\sum_{i=1}^N(y_i+40)}{N}=\frac{\sum_{i=1}^N(x_i+40)}{N}=\frac{\sum_{i=1}^Nx_i+N*40}{N}=E(X)+40=\mu_X+40$$` -- `$$\scriptsize \sigma_Y=\sqrt{Var(Y)}=\sqrt{\frac{\sum_i(y_i-\mu_y)^2}{N}}=\sqrt{\frac{\sum_i((x_i+40)-(\mu_x+40))^2}{N}}=\sqrt{\frac{\sum_i(x_i-\mu_x)^2}{N}}=\sqrt{Var(X)}=\sigma_X$$` `$$\scriptsize E(X+c)=E(X)+c \quad\text{and} \quad Var(X+c)=Var(X)$$` --- ### Standard Deviation - What happens to mean and standard deviation if everyone followers get multiplied by 3? (without addition) - `\(Y=3*X\)` <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/5aagf-1.png" width="100%" /> -- `$$\scriptsize \mu_Y=E(Y)=\frac{\sum_iy_i}{N}=\frac{\sum_i3x_i}{N}=3\frac{\sum_ix_i}{N}=3E(X)=3\mu_X$$` -- `$$\scriptsize \sigma_Y=\sqrt{Var(Y)}=\sqrt{\frac{\sum_i(y_i-\mu_y)^2}{N}}=\sqrt{\frac{\sum_i(3x_i-3\mu_x)^2}{N}}=\sqrt{3^2\frac{\sum_i(x_i-\mu_x)^2}{N}}=\sqrt{3^2Var(X)}=3\sigma_X$$` -- `$$\scriptsize E(cX)=cE(X) \quad\text{and} \quad Var(cX)=c^2Var(X)$$` --- ### Coefficient of Variation **Coefficient of Variation** divides the standard deviation by the mean. `$$C.V.=\frac{\sigma}{|\mu|}$$` And sample eqiuvalent `$$c.v.=\frac{s}{|\bar{x}|}$$` -- - Why? - It expresses standard deviation as proportion of the mean - Small value means variation is low compared to the mean - It is unit free - You can compare it across variables with different units/magnitudes --- ### Coefficient of Variation **Example** - variation of stocks in different currencies .pull-left[ <div class="datatables html-widget html-fill-item" id="htmlwidget-38d4e722b90c61f6e328" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-38d4e722b90c61f6e328">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["2023-07-01","2023-07-02","2023-07-03","2023-07-04","2023-07-05","2023-07-06","2023-07-07","2023-07-08","2023-07-09","2023-07-10","2023-07-11","2023-07-12","2023-07-13","2023-07-14","2023-07-15","2023-07-16","2023-07-17","2023-07-18","2023-07-19","2023-07-20"],[91.59,96.55,123.38,101.06,101.94,125.73,106.91,81.02,89.7,93.31999999999999,118.36,105.4,106.01,101.66,91.66,126.8,107.47,70.5,110.52,92.91],[1.01,1.16,1.02,1.07,1.09,0.9,1.35,1.23,1,1.43,1.28,1.15,1.36,1.36,1.35,1.32,1.3,1.19,1.14,1.13]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Date<\/th>\n <th>MXN_Stock<\/th>\n <th>USD_Stock<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]},{"name":"Date","targets":0},{"name":"MXN_Stock","targets":1},{"name":"USD_Stock","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> ] .pull-right[ - **Standard deviation**: - USD: 0.149 - MXN: 14.59 - **Coefficient of variation**: - USD: 0.12 - MXN: 0.14 ] --- ### Coefficient of Variation So more generally, if `\(y_i=bx_i\)`, then `$$C.V._y=\frac{\sigma_y}{|\mu_y|}=\frac{|b|\sigma_x}{|b\mu_x|}=C.V._x$$` What if `\(y_i=bx_i+a\)`? -- Then `$$C.V._y=\frac{\sigma_y}{|\mu_y|}=\frac{|b|\sigma_x}{|b\mu_x+a|} \neq C.V._x$$` --- ### Box and Whiskers plot - Helps to see the distribution of the data - Helps to see to see the outliers - Outliers are useful to see anomalies and potential errors in data collection - Whisker can be maximally 1.5 times the interquartile range - Any point beyond that is an outlier - If no point beyond 1.5 times the interquartile range, whisker goes just to the last datapoint and is shorter than 1.5 times the interquartile range <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/7a-1.png" width="100%" /> --- ### Box and Whiskers plot **Dataset comparisons** - They summarize data very well <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/8a-1.png" width="100%" /> --- ### Box and Whiskers plot **Dataset comparisons** - They summarize data very well <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/8b-1.png" width="100%" /> --- <center> <img src=Exam_q_bw.png width="800"> </center> ---